Cross-spectrum

In time series analysis, the cross-spectrum is used as part of a frequency domain analysis of the cross-correlation or cross-covariance between two time series.

Definition
Let $$(X_t,Y_t)$$ represent a pair of stochastic processes that are jointly wide sense stationary with autocovariance functions $$\gamma_{xx}$$ and $$\gamma_{yy}$$ and cross-covariance function $$\gamma_{xy}$$. Then the cross-spectrum $$\Gamma_{xy}$$ is defined as the Fourier transform of $$\gamma_{xy}$$



\Gamma_{xy}(f)= \mathcal{F}\{\gamma_{xy}\}(f) = \sum_{\tau=-\infty}^\infty \,\gamma_{xy}(\tau) \,e^{-2\,\pi\,i\,\tau\,f} , $$ where
 * $$\gamma_{xy}(\tau) = \operatorname{E}[(x_t - \mu_x)(y_{t+\tau} - \mu_y)]$$.

The cross-spectrum has representations as a decomposition into (i) its real part (co-spectrum) and (ii) its imaginary part (quadrature spectrum)

\Gamma_{xy}(f)= \Lambda_{xy}(f) - i \Psi_{xy}(f) , $$

and (ii) in polar coordinates

\Gamma_{xy}(f)= A_{xy}(f) \,e^{i \phi_{xy}(f) }. $$ Here, the amplitude spectrum $$A_{xy}$$ is given by
 * $$A_{xy}(f)= (\Lambda_{xy}(f)^2 + \Psi_{xy}(f)^2)^\frac{1}{2} ,$$

and the phase spectrum $$\Phi_{xy}$$ is given by
 * $$\begin{cases}

\tan^{-1} ( \Psi_{xy}(f) / \Lambda_{xy}(f)  )     & \text{if } \Psi_{xy}(f) \ne 0 \text{ and } \Lambda_{xy}(f) \ne 0 \\ 0    & \text{if } \Psi_{xy}(f) = 0 \text{ and } \Lambda_{xy}(f) > 0 \\ \pm \pi & \text{if } \Psi_{xy}(f) = 0 \text{ and } \Lambda_{xy}(f) < 0 \\ \pi/2 & \text{if } \Psi_{xy}(f) > 0 \text{ and } \Lambda_{xy}(f) = 0 \\ -\pi/2 & \text{if } \Psi_{xy}(f) < 0 \text{ and } \Lambda_{xy}(f) = 0 \\ \end{cases}$$

Squared coherency spectrum
The squared coherency spectrum is given by

\kappa_{xy}(f)= \frac{A_{xy}^2}{ \Gamma_{xx}(f) \Gamma_{yy}(f)} , $$

which expresses the amplitude spectrum in dimensionless units.